LMFDB - Embedded newform 1088.4.a.bg.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1088,4,Mod(1,1088)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1088, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1088.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1088 = 2^{6} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1088.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$64.1940780862$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 67x^{5} - 35x^{4} + 893x^{3} + 595x^{2} - 3064x - 2804 $ x^7 - x^6 - 67x^5 - 35x^4 + 893x^3 + 595x^2 - 3064*x - 2804
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{11} $
Twist minimal:	no (minimal twist has level 544)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.00457$ of defining polynomial
Character	$\chi$	$=$	1088.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.24805 q^{3} -3.51829 q^{5} +10.9586 q^{7} -16.4502 q^{9} +O(q^{10})$	$q+3.24805 q^{3} -3.51829 q^{5} +10.9586 q^{7} -16.4502 q^{9} -23.7437 q^{11} -47.1585 q^{13} -11.4276 q^{15} -17.0000 q^{17} +67.6749 q^{19} +35.5940 q^{21} +55.1122 q^{23} -112.622 q^{25} -141.128 q^{27} +237.588 q^{29} +303.801 q^{31} -77.1206 q^{33} -38.5555 q^{35} -122.719 q^{37} -153.173 q^{39} +435.595 q^{41} +232.067 q^{43} +57.8765 q^{45} +594.099 q^{47} -222.909 q^{49} -55.2168 q^{51} +392.312 q^{53} +83.5372 q^{55} +219.811 q^{57} -174.029 q^{59} -332.321 q^{61} -180.271 q^{63} +165.917 q^{65} -165.594 q^{67} +179.007 q^{69} -76.2120 q^{71} +559.023 q^{73} -365.800 q^{75} -260.198 q^{77} +1003.29 q^{79} -14.2359 q^{81} -1333.13 q^{83} +59.8109 q^{85} +771.696 q^{87} -31.9895 q^{89} -516.791 q^{91} +986.760 q^{93} -238.100 q^{95} +1633.36 q^{97} +390.588 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 2 q^{7} + 67 q^{9}+O(q^{10}) $ 7 * q - 2 * q^7 + 67 * q^9	$ 7 q - 2 q^{7} + 67 q^{9} - 108 q^{11} + 34 q^{13} + 128 q^{15} - 119 q^{17} - 124 q^{19} + 296 q^{21} - 6 q^{23} + 197 q^{25} - 248 q^{29} + 50 q^{31} + 512 q^{33} - 640 q^{35} + 484 q^{37} + 1504 q^{39} - 366 q^{41} - 1412 q^{43} + 80 q^{45} + 1012 q^{47} + 1115 q^{49} + 146 q^{53} + 1024 q^{55} + 48 q^{57} - 2332 q^{59} + 548 q^{61} + 2838 q^{63} - 208 q^{65} - 924 q^{67} + 1672 q^{69} + 1286 q^{71} + 870 q^{73} - 3136 q^{75} + 1344 q^{77} + 1818 q^{79} + 3039 q^{81} - 1772 q^{83} + 384 q^{87} + 1706 q^{89} - 588 q^{91} + 5576 q^{93} + 2048 q^{95} + 1802 q^{97} - 5148 q^{99}+O(q^{100}) $ 7 * q - 2 * q^7 + 67 * q^9 - 108 * q^11 + 34 * q^13 + 128 * q^15 - 119 * q^17 - 124 * q^19 + 296 * q^21 - 6 * q^23 + 197 * q^25 - 248 * q^29 + 50 * q^31 + 512 * q^33 - 640 * q^35 + 484 * q^37 + 1504 * q^39 - 366 * q^41 - 1412 * q^43 + 80 * q^45 + 1012 * q^47 + 1115 * q^49 + 146 * q^53 + 1024 * q^55 + 48 * q^57 - 2332 * q^59 + 548 * q^61 + 2838 * q^63 - 208 * q^65 - 924 * q^67 + 1672 * q^69 + 1286 * q^71 + 870 * q^73 - 3136 * q^75 + 1344 * q^77 + 1818 * q^79 + 3039 * q^81 - 1772 * q^83 + 384 * q^87 + 1706 * q^89 - 588 * q^91 + 5576 * q^93 + 2048 * q^95 + 1802 * q^97 - 5148 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	3.24805	0.625087	0.312543	−	0.949903i	$-0.398819\pi$
$3$	3.24805	0.625087	0.312543	+	0.949903i	$0.398819\pi$
$4$	0	0
$5$	−3.51829	−0.314685	−0.157343	−	0.987544i	$-0.550293\pi$
$5$	−3.51829	−0.314685	−0.157343	+	0.987544i	$0.550293\pi$
$6$	0	0
$7$	10.9586	0.591709	0.295854	−	0.955233i	$-0.404396\pi$
$7$	10.9586	0.591709	0.295854	+	0.955233i	$0.404396\pi$
$8$	0	0
$9$	−16.4502	−0.609266
$10$	0	0
$11$	−23.7437	−0.650818	−0.325409	−	0.945573i	$-0.605502\pi$
$11$	−23.7437	−0.650818	−0.325409	+	0.945573i	$0.605502\pi$
$12$	0	0
$13$	−47.1585	−1.00611	−0.503055	−	0.864255i	$-0.667791\pi$
$13$	−47.1585	−1.00611	−0.503055	+	0.864255i	$0.667791\pi$
$14$	0	0
$15$	−11.4276	−0.196706
$16$	0	0
$17$	−17.0000	−0.242536
$17$	−17.0000	−0.242536
$18$	0	0
$19$	67.6749	0.817141	0.408571	−	0.912727i	$-0.366027\pi$
$19$	67.6749	0.817141	0.408571	+	0.912727i	$0.366027\pi$
$20$	0	0
$21$	35.5940	0.369869
$22$	0	0
$23$	55.1122	0.499639	0.249820	−	0.968292i	$-0.419629\pi$
$23$	55.1122	0.499639	0.249820	+	0.968292i	$0.419629\pi$
$24$	0	0
$25$	−112.622	−0.900973
$26$	0	0
$27$	−141.128	−1.00593
$28$	0	0
$29$	237.588	1.52134	0.760672	−	0.649137i	$-0.224870\pi$
$29$	237.588	1.52134	0.760672	+	0.649137i	$0.224870\pi$
$30$	0	0
$31$	303.801	1.76014	0.880069	−	0.474846i	$-0.157496\pi$
$31$	303.801	1.76014	0.880069	+	0.474846i	$0.157496\pi$
$32$	0	0
$33$	−77.1206	−0.406818
$34$	0	0
$35$	−38.5555	−0.186202
$36$	0	0
$37$	−122.719	−0.545269	−0.272635	−	0.962118i	$-0.587895\pi$
$37$	−122.719	−0.545269	−0.272635	+	0.962118i	$0.587895\pi$
$38$	0	0
$39$	−153.173	−0.628906
$40$	0	0
$41$	435.595	1.65923	0.829615	−	0.558335i	$-0.188560\pi$
$41$	435.595	1.65923	0.829615	+	0.558335i	$0.188560\pi$
$42$	0	0
$43$	232.067	0.823020	0.411510	−	0.911405i	$-0.365002\pi$
$43$	232.067	0.823020	0.411510	+	0.911405i	$0.365002\pi$
$44$	0	0
$45$	57.8765	0.191727
$46$	0	0
$47$	594.099	1.84379	0.921897	−	0.387435i	$-0.126639\pi$
$47$	594.099	1.84379	0.921897	+	0.387435i	$0.126639\pi$
$48$	0	0
$49$	−222.909	−0.649881
$50$	0	0
$51$	−55.2168	−0.151606
$52$	0	0
$53$	392.312	1.01676	0.508379	−	0.861133i	$-0.330245\pi$
$53$	392.312	1.01676	0.508379	+	0.861133i	$0.330245\pi$
$54$	0	0
$55$	83.5372	0.204803
$56$	0	0
$57$	219.811	0.510784
$58$	0	0
$59$	−174.029	−0.384012	−0.192006	−	0.981394i	$-0.561499\pi$
$59$	−174.029	−0.384012	−0.192006	+	0.981394i	$0.561499\pi$
$60$	0	0
$61$	−332.321	−0.697530	−0.348765	−	0.937210i	$-0.613399\pi$
$61$	−332.321	−0.697530	−0.348765	+	0.937210i	$0.613399\pi$
$62$	0	0
$63$	−180.271	−0.360508
$64$	0	0
$65$	165.917	0.316608
$66$	0	0
$67$	−165.594	−0.301949	−0.150974	−	0.988538i	$-0.548241\pi$
$67$	−165.594	−0.301949	−0.150974	+	0.988538i	$0.548241\pi$
$68$	0	0
$69$	179.007	0.312318
$70$	0	0
$71$	−76.2120	−0.127390	−0.0636951	−	0.997969i	$-0.520289\pi$
$71$	−76.2120	−0.127390	−0.0636951	+	0.997969i	$0.520289\pi$
$72$	0	0
$73$	559.023	0.896284	0.448142	−	0.893962i	$-0.352086\pi$
$73$	559.023	0.896284	0.448142	+	0.893962i	$0.352086\pi$
$74$	0	0
$75$	−365.800	−0.563187
$76$	0	0
$77$	−260.198	−0.385094
$78$	0	0
$79$	1003.29	1.42885	0.714423	−	0.699714i	$-0.246689\pi$
$79$	1003.29	1.42885	0.714423	+	0.699714i	$0.246689\pi$
$80$	0	0
$81$	−14.2359	−0.0195280
$82$	0	0
$83$	−1333.13	−1.76301	−0.881504	−	0.472177i	$-0.843468\pi$
$83$	−1333.13	−1.76301	−0.881504	+	0.472177i	$0.843468\pi$
$84$	0	0
$85$	59.8109	0.0763224
$86$	0	0
$87$	771.696	0.950972
$88$	0	0
$89$	−31.9895	−0.0380998	−0.0190499	−	0.999819i	$-0.506064\pi$
$89$	−31.9895	−0.0380998	−0.0190499	+	0.999819i	$0.506064\pi$
$90$	0	0
$91$	−516.791	−0.595323
$92$	0	0
$93$	986.760	1.10024
$94$	0	0
$95$	−238.100	−0.257142
$96$	0	0
$97$	1633.36	1.70972	0.854859	−	0.518860i	$-0.173643\pi$
$97$	1633.36	1.70972	0.854859	+	0.518860i	$0.173643\pi$
$98$	0	0
$99$	390.588	0.396521
$100$	0	0
$101$	−1019.74	−1.00463	−0.502316	−	0.864684i	$-0.667519\pi$
$101$	−1019.74	−1.00463	−0.502316	+	0.864684i	$0.667519\pi$
$102$	0	0
$103$	754.337	0.721622	0.360811	−	0.932639i	$-0.382500\pi$
$103$	754.337	0.721622	0.360811	+	0.932639i	$0.382500\pi$
$104$	0	0
$105$	−125.230	−0.116392
$106$	0	0
$107$	−1165.48	−1.05300	−0.526501	−	0.850175i	$-0.676496\pi$
$107$	−1165.48	−1.05300	−0.526501	+	0.850175i	$0.676496\pi$
$108$	0	0
$109$	1241.76	1.09118	0.545592	−	0.838051i	$-0.316305\pi$
$109$	1241.76	1.09118	0.545592	+	0.838051i	$0.316305\pi$
$110$	0	0
$111$	−398.598	−0.340840
$112$	0	0
$113$	262.115	0.218209	0.109105	−	0.994030i	$-0.465202\pi$
$113$	262.115	0.218209	0.109105	+	0.994030i	$0.465202\pi$
$114$	0	0
$115$	−193.901	−0.157229
$116$	0	0
$117$	775.767	0.612989
$118$	0	0
$119$	−186.296	−0.143510
$120$	0	0
$121$	−767.237	−0.576436
$122$	0	0
$123$	1414.83	1.03716
$124$	0	0
$125$	836.021	0.598208
$126$	0	0
$127$	722.607	0.504890	0.252445	−	0.967611i	$-0.418765\pi$
$127$	722.607	0.504890	0.252445	+	0.967611i	$0.418765\pi$
$128$	0	0
$129$	753.764	0.514459
$130$	0	0
$131$	1673.19	1.11593	0.557966	−	0.829864i	$-0.311582\pi$
$131$	1673.19	1.11593	0.557966	+	0.829864i	$0.311582\pi$
$132$	0	0
$133$	741.622	0.483509
$134$	0	0
$135$	496.530	0.316552
$136$	0	0
$137$	2338.46	1.45831	0.729154	−	0.684349i	$-0.239914\pi$
$137$	2338.46	1.45831	0.729154	+	0.684349i	$0.239914\pi$
$138$	0	0
$139$	−263.184	−0.160597	−0.0802985	−	0.996771i	$-0.525587\pi$
$139$	−263.184	−0.160597	−0.0802985	+	0.996771i	$0.525587\pi$
$140$	0	0
$141$	1929.66	1.15253
$142$	0	0
$143$	1119.72	0.654794
$144$	0	0
$145$	−835.902	−0.478744
$146$	0	0
$147$	−724.019	−0.406232
$148$	0	0
$149$	−1430.79	−0.786676	−0.393338	−	0.919394i	$-0.628680\pi$
$149$	−1430.79	−0.786676	−0.393338	+	0.919394i	$0.628680\pi$
$150$	0	0
$151$	−2818.89	−1.51919	−0.759596	−	0.650395i	$-0.774604\pi$
$151$	−2818.89	−1.51919	−0.759596	+	0.650395i	$0.774604\pi$
$152$	0	0
$153$	279.653	0.147769
$154$	0	0
$155$	−1068.86	−0.553890
$156$	0	0
$157$	2466.38	1.25375	0.626874	−	0.779121i	$-0.284334\pi$
$157$	2466.38	1.25375	0.626874	+	0.779121i	$0.284334\pi$
$158$	0	0
$159$	1274.25	0.635562
$160$	0	0
$161$	603.953	0.295641
$162$	0	0
$163$	−865.510	−0.415902	−0.207951	−	0.978139i	$-0.566679\pi$
$163$	−865.510	−0.415902	−0.207951	+	0.978139i	$0.566679\pi$
$164$	0	0
$165$	271.333	0.128019
$166$	0	0
$167$	−2024.39	−0.938036	−0.469018	−	0.883189i	$-0.655392\pi$
$167$	−2024.39	−0.938036	−0.469018	+	0.883189i	$0.655392\pi$
$168$	0	0
$169$	26.9258	0.0122557
$170$	0	0
$171$	−1113.26	−0.497857
$172$	0	0
$173$	663.819	0.291729	0.145865	−	0.989305i	$-0.453404\pi$
$173$	663.819	0.291729	0.145865	+	0.989305i	$0.453404\pi$
$174$	0	0
$175$	−1234.18	−0.533114
$176$	0	0
$177$	−565.256	−0.240041
$178$	0	0
$179$	70.2472	0.0293325	0.0146663	−	0.999892i	$-0.495331\pi$
$179$	70.2472	0.0293325	0.0146663	+	0.999892i	$0.495331\pi$
$180$	0	0
$181$	−926.748	−0.380578	−0.190289	−	0.981728i	$-0.560943\pi$
$181$	−926.748	−0.380578	−0.190289	+	0.981728i	$0.560943\pi$
$182$	0	0
$183$	−1079.39	−0.436017
$184$	0	0
$185$	431.762	0.171588
$186$	0	0
$187$	403.643	0.157846
$188$	0	0
$189$	−1546.57	−0.595218
$190$	0	0
$191$	−1970.62	−0.746540	−0.373270	−	0.927723i	$-0.621764\pi$
$191$	−1970.62	−0.746540	−0.373270	+	0.927723i	$0.621764\pi$
$192$	0	0
$193$	2101.01	0.783597	0.391799	−	0.920051i	$-0.371853\pi$
$193$	2101.01	0.783597	0.391799	+	0.920051i	$0.371853\pi$
$194$	0	0
$195$	538.907	0.197907
$196$	0	0
$197$	3671.99	1.32801	0.664006	−	0.747727i	$-0.268855\pi$
$197$	3671.99	1.32801	0.664006	+	0.747727i	$0.268855\pi$
$198$	0	0
$199$	1161.43	0.413726	0.206863	−	0.978370i	$-0.433675\pi$
$199$	1161.43	0.413726	0.206863	+	0.978370i	$0.433675\pi$
$200$	0	0
$201$	−537.858	−0.188744
$202$	0	0
$203$	2603.63	0.900192
$204$	0	0
$205$	−1532.55	−0.522135
$206$	0	0
$207$	−906.607	−0.304413
$208$	0	0
$209$	−1606.85	−0.531810
$210$	0	0
$211$	3709.86	1.21041	0.605207	−	0.796068i	$-0.293090\pi$
$211$	3709.86	1.21041	0.605207	+	0.796068i	$0.293090\pi$
$212$	0	0
$213$	−247.540	−0.0796299
$214$	0	0
$215$	−816.478	−0.258992
$216$	0	0
$217$	3329.23	1.04149
$218$	0	0
$219$	1815.73	0.560255
$220$	0	0
$221$	801.695	0.244017
$222$	0	0
$223$	−397.023	−0.119223	−0.0596113	−	0.998222i	$-0.518986\pi$
$223$	−397.023	−0.119223	−0.0596113	+	0.998222i	$0.518986\pi$
$224$	0	0
$225$	1852.65	0.548933
$226$	0	0
$227$	−1730.14	−0.505875	−0.252938	−	0.967483i	$-0.581397\pi$
$227$	−1730.14	−0.505875	−0.252938	+	0.967483i	$0.581397\pi$
$228$	0	0
$229$	5103.01	1.47256	0.736280	−	0.676677i	$-0.236581\pi$
$229$	5103.01	1.47256	0.736280	+	0.676677i	$0.236581\pi$
$230$	0	0
$231$	−845.134	−0.240717
$232$	0	0
$233$	1961.49	0.551509	0.275755	−	0.961228i	$-0.411072\pi$
$233$	1961.49	0.551509	0.275755	+	0.961228i	$0.411072\pi$
$234$	0	0
$235$	−2090.21	−0.580215
$236$	0	0
$237$	3258.73	0.893153
$238$	0	0
$239$	−5798.95	−1.56947	−0.784734	−	0.619833i	$-0.787200\pi$
$239$	−5798.95	−1.56947	−0.784734	+	0.619833i	$0.787200\pi$
$240$	0	0
$241$	5483.62	1.46569	0.732845	−	0.680396i	$-0.238192\pi$
$241$	5483.62	1.46569	0.732845	+	0.680396i	$0.238192\pi$
$242$	0	0
$243$	3764.22	0.993725
$244$	0	0
$245$	784.259	0.204508
$246$	0	0
$247$	−3191.45	−0.822133
$248$	0	0
$249$	−4330.06	−1.10203
$250$	0	0
$251$	−3381.15	−0.850265	−0.425133	−	0.905131i	$-0.639773\pi$
$251$	−3381.15	−0.850265	−0.425133	+	0.905131i	$0.639773\pi$
$252$	0	0
$253$	−1308.57	−0.325174
$254$	0	0
$255$	194.269	0.0477081
$256$	0	0
$257$	−5482.55	−1.33071	−0.665355	−	0.746527i	$-0.731720\pi$
$257$	−5482.55	−1.33071	−0.665355	+	0.746527i	$0.731720\pi$
$258$	0	0
$259$	−1344.83	−0.322640
$260$	0	0
$261$	−3908.37	−0.926903
$262$	0	0
$263$	970.197	0.227471	0.113736	−	0.993511i	$-0.463718\pi$
$263$	970.197	0.227471	0.113736	+	0.993511i	$0.463718\pi$
$264$	0	0
$265$	−1380.27	−0.319959
$266$	0	0
$267$	−103.903	−0.0238157
$268$	0	0
$269$	−3255.29	−0.737839	−0.368919	−	0.929461i	$-0.620272\pi$
$269$	−3255.29	−0.737839	−0.368919	+	0.929461i	$0.620272\pi$
$270$	0	0
$271$	4122.65	0.924109	0.462054	−	0.886852i	$-0.347113\pi$
$271$	4122.65	0.924109	0.462054	+	0.886852i	$0.347113\pi$
$272$	0	0
$273$	−1678.56	−0.372129
$274$	0	0
$275$	2674.05	0.586369
$276$	0	0
$277$	−6463.78	−1.40206	−0.701031	−	0.713131i	$-0.747277\pi$
$277$	−6463.78	−1.40206	−0.701031	+	0.713131i	$0.747277\pi$
$278$	0	0
$279$	−4997.59	−1.07239
$280$	0	0
$281$	8077.95	1.71491	0.857456	−	0.514558i	$-0.172044\pi$
$281$	8077.95	1.71491	0.857456	+	0.514558i	$0.172044\pi$
$282$	0	0
$283$	8426.73	1.77003	0.885013	−	0.465566i	$-0.154149\pi$
$283$	8426.73	1.77003	0.885013	+	0.465566i	$0.154149\pi$
$284$	0	0
$285$	−773.359	−0.160736
$286$	0	0
$287$	4773.51	0.981781
$288$	0	0
$289$	289.000	0.0588235
$290$	0	0
$291$	5305.23	1.06872
$292$	0	0
$293$	−1742.82	−0.347496	−0.173748	−	0.984790i	$-0.555588\pi$
$293$	−1742.82	−0.347496	−0.173748	+	0.984790i	$0.555588\pi$
$294$	0	0
$295$	612.286	0.120843
$296$	0	0
$297$	3350.91	0.654678
$298$	0	0
$299$	−2599.01	−0.502691
$300$	0	0
$301$	2543.13	0.486988
$302$	0	0
$303$	−3312.16	−0.627983
$304$	0	0
$305$	1169.20	0.219503
$306$	0	0
$307$	9097.79	1.69133	0.845665	−	0.533714i	$-0.179204\pi$
$307$	9097.79	1.69133	0.845665	+	0.533714i	$0.179204\pi$
$308$	0	0
$309$	2450.12	0.451076
$310$	0	0
$311$	8491.06	1.54818	0.774090	−	0.633076i	$-0.218208\pi$
$311$	8491.06	1.54818	0.774090	+	0.633076i	$0.218208\pi$
$312$	0	0
$313$	134.718	0.0243282	0.0121641	−	0.999926i	$-0.496128\pi$
$313$	134.718	0.0243282	0.0121641	+	0.999926i	$0.496128\pi$
$314$	0	0
$315$	634.245	0.113447
$316$	0	0
$317$	2901.67	0.514113	0.257056	−	0.966396i	$-0.417247\pi$
$317$	2901.67	0.514113	0.257056	+	0.966396i	$0.417247\pi$
$318$	0	0
$319$	−5641.21	−0.990117
$320$	0	0
$321$	−3785.53	−0.658218
$322$	0	0
$323$	−1150.47	−0.198186
$324$	0	0
$325$	5311.07	0.906477
$326$	0	0
$327$	4033.29	0.682085
$328$	0	0
$329$	6510.50	1.09099
$330$	0	0
$331$	1895.83	0.314816	0.157408	−	0.987534i	$-0.449686\pi$
$331$	1895.83	0.314816	0.157408	+	0.987534i	$0.449686\pi$
$332$	0	0
$333$	2018.76	0.332214
$334$	0	0
$335$	582.609	0.0950189
$336$	0	0
$337$	93.0976	0.0150485	0.00752426	−	0.999972i	$-0.497605\pi$
$337$	93.0976	0.0150485	0.00752426	+	0.999972i	$0.497605\pi$
$338$	0	0
$339$	851.360	0.136400
$340$	0	0
$341$	−7213.36	−1.14553
$342$	0	0
$343$	−6201.57	−0.976249
$344$	0	0
$345$	−629.799	−0.0982818
$346$	0	0
$347$	−7925.10	−1.22606	−0.613028	−	0.790061i	$-0.710049\pi$
$347$	−7925.10	−1.22606	−0.613028	+	0.790061i	$0.710049\pi$
$348$	0	0
$349$	4721.70	0.724202	0.362101	−	0.932139i	$-0.382059\pi$
$349$	4721.70	0.724202	0.362101	+	0.932139i	$0.382059\pi$
$350$	0	0
$351$	6655.40	1.01208
$352$	0	0
$353$	−11711.7	−1.76586	−0.882931	−	0.469503i	$-0.844433\pi$
$353$	−11711.7	−1.76586	−0.882931	+	0.469503i	$0.844433\pi$
$354$	0	0
$355$	268.136	0.0400878
$356$	0	0
$357$	−605.099	−0.0897065
$358$	0	0
$359$	−1754.65	−0.257958	−0.128979	−	0.991647i	$-0.541170\pi$
$359$	−1754.65	−0.257958	−0.128979	+	0.991647i	$0.541170\pi$
$360$	0	0
$361$	−2279.11	−0.332280
$362$	0	0
$363$	−2492.02	−0.360323
$364$	0	0
$365$	−1966.80	−0.282047
$366$	0	0
$367$	−6298.45	−0.895848	−0.447924	−	0.894072i	$-0.647836\pi$
$367$	−6298.45	−0.895848	−0.447924	+	0.894072i	$0.647836\pi$
$368$	0	0
$369$	−7165.62	−1.01091
$370$	0	0
$371$	4299.19	0.601625
$372$	0	0
$373$	−6682.68	−0.927657	−0.463828	−	0.885925i	$-0.653525\pi$
$373$	−6682.68	−0.927657	−0.463828	+	0.885925i	$0.653525\pi$
$374$	0	0
$375$	2715.44	0.373932
$376$	0	0
$377$	−11204.3	−1.53064
$378$	0	0
$379$	1058.29	0.143432	0.0717158	−	0.997425i	$-0.477153\pi$
$379$	1058.29	0.143432	0.0717158	+	0.997425i	$0.477153\pi$
$380$	0	0
$381$	2347.06	0.315600
$382$	0	0
$383$	9898.73	1.32063	0.660315	−	0.750989i	$-0.270423\pi$
$383$	9898.73	1.32063	0.660315	+	0.750989i	$0.270423\pi$
$384$	0	0
$385$	915.450	0.121184
$386$	0	0
$387$	−3817.54	−0.501438
$388$	0	0
$389$	−305.994	−0.0398831	−0.0199416	−	0.999801i	$-0.506348\pi$
$389$	−305.994	−0.0398831	−0.0199416	+	0.999801i	$0.506348\pi$
$390$	0	0
$391$	−936.908	−0.121180
$392$	0	0
$393$	5434.59	0.697554
$394$	0	0
$395$	−3529.86	−0.449637
$396$	0	0
$397$	4705.55	0.594874	0.297437	−	0.954741i	$-0.403868\pi$
$397$	4705.55	0.594874	0.297437	+	0.954741i	$0.403868\pi$
$398$	0	0
$399$	2408.82	0.302235
$400$	0	0
$401$	−7468.90	−0.930122	−0.465061	−	0.885279i	$-0.653968\pi$
$401$	−7468.90	−0.930122	−0.465061	+	0.885279i	$0.653968\pi$
$402$	0	0
$403$	−14326.8	−1.77089
$404$	0	0
$405$	50.0861	0.00614518
$406$	0	0
$407$	2913.81	0.354871
$408$	0	0
$409$	9585.99	1.15892	0.579458	−	0.815002i	$-0.303264\pi$
$409$	9585.99	1.15892	0.579458	+	0.815002i	$0.303264\pi$
$410$	0	0
$411$	7595.43	0.911570
$412$	0	0
$413$	−1907.12	−0.227223
$414$	0	0
$415$	4690.32	0.554793
$416$	0	0
$417$	−854.834	−0.100387
$418$	0	0
$419$	−196.371	−0.0228958	−0.0114479	−	0.999934i	$-0.503644\pi$
$419$	−196.371	−0.0228958	−0.0114479	+	0.999934i	$0.503644\pi$
$420$	0	0
$421$	−12431.5	−1.43914	−0.719568	−	0.694421i	$-0.755660\pi$
$421$	−12431.5	−1.43914	−0.719568	+	0.694421i	$0.755660\pi$
$422$	0	0
$423$	−9773.05	−1.12336
$424$	0	0
$425$	1914.57	0.218518
$426$	0	0
$427$	−3641.77	−0.412735
$428$	0	0
$429$	3636.89	0.409303
$430$	0	0
$431$	6717.95	0.750794	0.375397	−	0.926864i	$-0.377506\pi$
$431$	6717.95	0.750794	0.375397	+	0.926864i	$0.377506\pi$
$432$	0	0
$433$	−11135.8	−1.23592	−0.617960	−	0.786209i	$-0.712041\pi$
$433$	−11135.8	−1.23592	−0.617960	+	0.786209i	$0.712041\pi$
$434$	0	0
$435$	−2715.05	−0.299257
$436$	0	0
$437$	3729.71	0.408276
$438$	0	0
$439$	7516.31	0.817162	0.408581	−	0.912722i	$-0.366024\pi$
$439$	7516.31	0.817162	0.408581	+	0.912722i	$0.366024\pi$
$440$	0	0
$441$	3666.90	0.395951
$442$	0	0
$443$	−1252.05	−0.134282	−0.0671409	−	0.997744i	$-0.521388\pi$
$443$	−1252.05	−0.134282	−0.0671409	+	0.997744i	$0.521388\pi$
$444$	0	0
$445$	112.548	0.0119894
$446$	0	0
$447$	−4647.27	−0.491741
$448$	0	0
$449$	−5794.88	−0.609081	−0.304540	−	0.952499i	$-0.598503\pi$
$449$	−5794.88	−0.609081	−0.304540	+	0.952499i	$0.598503\pi$
$450$	0	0
$451$	−10342.6	−1.07986
$452$	0	0
$453$	−9155.89	−0.949628
$454$	0	0
$455$	1818.22	0.187340
$456$	0	0
$457$	−1188.56	−0.121660	−0.0608298	−	0.998148i	$-0.519375\pi$
$457$	−1188.56	−0.121660	−0.0608298	+	0.998148i	$0.519375\pi$
$458$	0	0
$459$	2399.18	0.243974
$460$	0	0
$461$	−6391.87	−0.645768	−0.322884	−	0.946439i	$-0.604652\pi$
$461$	−6391.87	−0.645768	−0.322884	+	0.946439i	$0.604652\pi$
$462$	0	0
$463$	−8214.34	−0.824520	−0.412260	−	0.911066i	$-0.635260\pi$
$463$	−8214.34	−0.824520	−0.412260	+	0.911066i	$0.635260\pi$
$464$	0	0
$465$	−3471.71	−0.346229
$466$	0	0
$467$	−16444.2	−1.62944	−0.814718	−	0.579857i	$-0.803108\pi$
$467$	−16444.2	−1.62944	−0.814718	+	0.579857i	$0.803108\pi$
$468$	0	0
$469$	−1814.68	−0.178666
$470$	0	0
$471$	8010.91	0.783701
$472$	0	0
$473$	−5510.12	−0.535636
$474$	0	0
$475$	−7621.66	−0.736222
$476$	0	0
$477$	−6453.61	−0.619477
$478$	0	0
$479$	10295.6	0.982082	0.491041	−	0.871137i	$-0.336617\pi$
$479$	10295.6	0.982082	0.491041	+	0.871137i	$0.336617\pi$
$480$	0	0
$481$	5787.27	0.548600
$482$	0	0
$483$	1961.67	0.184801
$484$	0	0
$485$	−5746.63	−0.538023
$486$	0	0
$487$	−773.861	−0.0720061	−0.0360031	−	0.999352i	$-0.511463\pi$
$487$	−773.861	−0.0720061	−0.0360031	+	0.999352i	$0.511463\pi$
$488$	0	0
$489$	−2811.22	−0.259975
$490$	0	0
$491$	−7715.06	−0.709115	−0.354558	−	0.935034i	$-0.615369\pi$
$491$	−7715.06	−0.709115	−0.354558	+	0.935034i	$0.615369\pi$
$492$	0	0
$493$	−4038.99	−0.368980
$494$	0	0
$495$	−1374.20	−0.124779
$496$	0	0
$497$	−835.177	−0.0753779
$498$	0	0
$499$	3926.92	0.352291	0.176146	−	0.984364i	$-0.443637\pi$
$499$	3926.92	0.352291	0.176146	+	0.984364i	$0.443637\pi$
$500$	0	0
$501$	−6575.31	−0.586354
$502$	0	0
$503$	−3593.23	−0.318517	−0.159258	−	0.987237i	$-0.550910\pi$
$503$	−3593.23	−0.318517	−0.159258	+	0.987237i	$0.550910\pi$
$504$	0	0
$505$	3587.74	0.316143
$506$	0	0
$507$	87.4563	0.00766089
$508$	0	0
$509$	−6533.93	−0.568981	−0.284490	−	0.958679i	$-0.591824\pi$
$509$	−6533.93	−0.568981	−0.284490	+	0.958679i	$0.591824\pi$
$510$	0	0
$511$	6126.11	0.530339
$512$	0	0
$513$	−9550.84	−0.821988
$514$	0	0
$515$	−2653.98	−0.227084
$516$	0	0
$517$	−14106.1	−1.19997
$518$	0	0
$519$	2156.11	0.182356
$520$	0	0
$521$	−12512.0	−1.05213	−0.526066	−	0.850444i	$-0.676334\pi$
$521$	−12512.0	−1.05213	−0.526066	+	0.850444i	$0.676334\pi$
$522$	0	0
$523$	−10715.0	−0.895863	−0.447932	−	0.894068i	$-0.647839\pi$
$523$	−10715.0	−0.895863	−0.447932	+	0.894068i	$0.647839\pi$
$524$	0	0
$525$	−4008.66	−0.333242
$526$	0	0
$527$	−5164.62	−0.426896
$528$	0	0
$529$	−9129.64	−0.750361
$530$	0	0
$531$	2862.82	0.233966
$532$	0	0
$533$	−20542.0	−1.66937
$534$	0	0
$535$	4100.49	0.331364
$536$	0	0
$537$	228.166	0.0183354
$538$	0	0
$539$	5292.69	0.422954
$540$	0	0
$541$	20006.9	1.58995	0.794975	−	0.606642i	$-0.207484\pi$
$541$	20006.9	1.58995	0.794975	+	0.606642i	$0.207484\pi$
$542$	0	0
$543$	−3010.12	−0.237894
$544$	0	0
$545$	−4368.87	−0.343379
$546$	0	0
$547$	16250.5	1.27024	0.635121	−	0.772413i	$-0.280950\pi$
$547$	16250.5	1.27024	0.635121	+	0.772413i	$0.280950\pi$
$548$	0	0
$549$	5466.75	0.424982
$550$	0	0
$551$	16078.7	1.24315
$552$	0	0
$553$	10994.6	0.845460
$554$	0	0
$555$	1402.38	0.107257
$556$	0	0
$557$	8475.62	0.644746	0.322373	−	0.946613i	$-0.395519\pi$
$557$	8475.62	0.644746	0.322373	+	0.946613i	$0.395519\pi$
$558$	0	0
$559$	−10943.9	−0.828048
$560$	0	0
$561$	1311.05	0.0986677
$562$	0	0
$563$	−22563.0	−1.68902	−0.844508	−	0.535543i	$-0.820107\pi$
$563$	−22563.0	−1.68902	−0.844508	+	0.535543i	$0.820107\pi$
$564$	0	0
$565$	−922.194	−0.0686673
$566$	0	0
$567$	−156.006	−0.0115549
$568$	0	0
$569$	1820.31	0.134115	0.0670576	−	0.997749i	$-0.478639\pi$
$569$	1820.31	0.134115	0.0670576	+	0.997749i	$0.478639\pi$
$570$	0	0
$571$	12294.2	0.901043	0.450522	−	0.892766i	$-0.351238\pi$
$571$	12294.2	0.901043	0.450522	+	0.892766i	$0.351238\pi$
$572$	0	0
$573$	−6400.67	−0.466652
$574$	0	0
$575$	−6206.83	−0.450161
$576$	0	0
$577$	−10948.3	−0.789918	−0.394959	−	0.918699i	$-0.629241\pi$
$577$	−10948.3	−0.789918	−0.394959	+	0.918699i	$0.629241\pi$
$578$	0	0
$579$	6824.19	0.489816
$580$	0	0
$581$	−14609.2	−1.04319
$582$	0	0
$583$	−9314.94	−0.661724
$584$	0	0
$585$	−2729.37	−0.192898
$586$	0	0
$587$	5593.92	0.393332	0.196666	−	0.980471i	$-0.436989\pi$
$587$	5593.92	0.393332	0.196666	+	0.980471i	$0.436989\pi$
$588$	0	0
$589$	20559.7	1.43828
$590$	0	0
$591$	11926.8	0.830123
$592$	0	0
$593$	15467.3	1.07111	0.535554	−	0.844501i	$-0.320103\pi$
$593$	15467.3	1.07111	0.535554	+	0.844501i	$0.320103\pi$
$594$	0	0
$595$	655.443	0.0451606
$596$	0	0
$597$	3772.37	0.258615
$598$	0	0
$599$	20969.4	1.43036	0.715181	−	0.698939i	$-0.246344\pi$
$599$	20969.4	1.43036	0.715181	+	0.698939i	$0.246344\pi$
$600$	0	0
$601$	−10747.5	−0.729447	−0.364724	−	0.931116i	$-0.618837\pi$
$601$	−10747.5	−0.729447	−0.364724	+	0.931116i	$0.618837\pi$
$602$	0	0
$603$	2724.06	0.183967
$604$	0	0
$605$	2699.36	0.181396
$606$	0	0
$607$	−12074.0	−0.807364	−0.403682	−	0.914899i	$-0.632270\pi$
$607$	−12074.0	−0.807364	−0.403682	+	0.914899i	$0.632270\pi$
$608$	0	0
$609$	8456.71	0.562698
$610$	0	0
$611$	−28016.9	−1.85506
$612$	0	0
$613$	9438.87	0.621912	0.310956	−	0.950424i	$-0.399351\pi$
$613$	9438.87	0.621912	0.310956	+	0.950424i	$0.399351\pi$
$614$	0	0
$615$	−4977.79	−0.326380
$616$	0	0
$617$	18221.6	1.18893	0.594467	−	0.804120i	$-0.297363\pi$
$617$	18221.6	1.18893	0.594467	+	0.804120i	$0.297363\pi$
$618$	0	0
$619$	−12655.7	−0.821772	−0.410886	−	0.911687i	$-0.634780\pi$
$619$	−12655.7	−0.821772	−0.410886	+	0.911687i	$0.634780\pi$
$620$	0	0
$621$	−7777.90	−0.502603
$622$	0	0
$623$	−350.560	−0.0225440
$624$	0	0
$625$	11136.3	0.712726
$626$	0	0
$627$	−5219.13	−0.332427
$628$	0	0
$629$	2086.23	0.132247
$630$	0	0
$631$	−12669.0	−0.799276	−0.399638	−	0.916673i	$-0.630864\pi$
$631$	−12669.0	−0.799276	−0.399638	+	0.916673i	$0.630864\pi$
$632$	0	0
$633$	12049.8	0.756614
$634$	0	0
$635$	−2542.34	−0.158881
$636$	0	0
$637$	10512.1	0.653851
$638$	0	0
$639$	1253.70	0.0776145
$640$	0	0
$641$	−11292.9	−0.695856	−0.347928	−	0.937521i	$-0.613115\pi$
$641$	−11292.9	−0.695856	−0.347928	+	0.937521i	$0.613115\pi$
$642$	0	0
$643$	−746.447	−0.0457807	−0.0228904	−	0.999738i	$-0.507287\pi$
$643$	−746.447	−0.0457807	−0.0228904	+	0.999738i	$0.507287\pi$
$644$	0	0
$645$	−2651.96	−0.161893
$646$	0	0
$647$	−17692.6	−1.07507	−0.537533	−	0.843243i	$-0.680644\pi$
$647$	−17692.6	−1.07507	−0.537533	+	0.843243i	$0.680644\pi$
$648$	0	0
$649$	4132.10	0.249922
$650$	0	0
$651$	10813.5	0.651021
$652$	0	0
$653$	−132.798	−0.00795833	−0.00397916	−	0.999992i	$-0.501267\pi$
$653$	−132.798	−0.00795833	−0.00397916	+	0.999992i	$0.501267\pi$
$654$	0	0
$655$	−5886.75	−0.351167
$656$	0	0
$657$	−9196.04	−0.546076
$658$	0	0
$659$	−17767.6	−1.05027	−0.525136	−	0.851018i	$-0.675985\pi$
$659$	−17767.6	−1.05027	−0.525136	+	0.851018i	$0.675985\pi$
$660$	0	0
$661$	−20422.1	−1.20171	−0.600854	−	0.799359i	$-0.705173\pi$
$661$	−20422.1	−1.20171	−0.600854	+	0.799359i	$0.705173\pi$
$662$	0	0
$663$	2603.94	0.152532
$664$	0	0
$665$	−2609.24	−0.152153
$666$	0	0
$667$	13094.0	0.760123
$668$	0	0
$669$	−1289.55	−0.0745245
$670$	0	0
$671$	7890.53	0.453965
$672$	0	0
$673$	−30616.6	−1.75361	−0.876807	−	0.480842i	$-0.840331\pi$
$673$	−30616.6	−1.75361	−0.876807	+	0.480842i	$0.840331\pi$
$674$	0	0
$675$	15894.1	0.906317
$676$	0	0
$677$	18997.9	1.07851	0.539253	−	0.842144i	$-0.318707\pi$
$677$	18997.9	1.07851	0.539253	+	0.842144i	$0.318707\pi$
$678$	0	0
$679$	17899.3	1.01166
$680$	0	0
$681$	−5619.59	−0.316216
$682$	0	0
$683$	33428.1	1.87275	0.936377	−	0.350996i	$-0.114157\pi$
$683$	33428.1	1.87275	0.936377	+	0.350996i	$0.114157\pi$
$684$	0	0
$685$	−8227.38	−0.458908
$686$	0	0
$687$	16574.8	0.920478
$688$	0	0
$689$	−18500.8	−1.02297
$690$	0	0
$691$	26903.4	1.48112	0.740560	−	0.671991i	$-0.234560\pi$
$691$	26903.4	1.48112	0.740560	+	0.671991i	$0.234560\pi$
$692$	0	0
$693$	4280.30	0.234625
$694$	0	0
$695$	925.958	0.0505375
$696$	0	0
$697$	−7405.11	−0.402423
$698$	0	0
$699$	6371.02	0.344741
$700$	0	0
$701$	26025.9	1.40226	0.701129	−	0.713034i	$-0.252680\pi$
$701$	26025.9	1.40226	0.701129	+	0.713034i	$0.252680\pi$
$702$	0	0
$703$	−8305.02	−0.445562
$704$	0	0
$705$	−6789.11	−0.362685
$706$	0	0
$707$	−11174.9	−0.594450
$708$	0	0
$709$	−1570.27	−0.0831774	−0.0415887	−	0.999135i	$-0.513242\pi$
$709$	−1570.27	−0.0831774	−0.0415887	+	0.999135i	$0.513242\pi$
$710$	0	0
$711$	−16504.3	−0.870548
$712$	0	0
$713$	16743.2	0.879434
$714$	0	0
$715$	−3939.49	−0.206054
$716$	0	0
$717$	−18835.2	−0.981053
$718$	0	0
$719$	−1539.79	−0.0798669	−0.0399335	−	0.999202i	$-0.512715\pi$
$719$	−1539.79	−0.0798669	−0.0399335	+	0.999202i	$0.512715\pi$
$720$	0	0
$721$	8266.48	0.426990
$722$	0	0
$723$	17811.1	0.916183
$724$	0	0
$725$	−26757.5	−1.37069
$726$	0	0
$727$	21003.5	1.07150	0.535748	−	0.844378i	$-0.320030\pi$
$727$	21003.5	1.07150	0.535748	+	0.844378i	$0.320030\pi$
$728$	0	0
$729$	12610.7	0.640692
$730$	0	0
$731$	−3945.14	−0.199612
$732$	0	0
$733$	14983.1	0.754997	0.377498	−	0.926010i	$-0.376784\pi$
$733$	14983.1	0.754997	0.377498	+	0.926010i	$0.376784\pi$
$734$	0	0
$735$	2547.31	0.127835
$736$	0	0
$737$	3931.82	0.196514
$738$	0	0
$739$	−2001.07	−0.0996082	−0.0498041	−	0.998759i	$-0.515860\pi$
$739$	−2001.07	−0.0996082	−0.0498041	+	0.998759i	$0.515860\pi$
$740$	0	0
$741$	−10366.0	−0.513905
$742$	0	0
$743$	−13139.2	−0.648765	−0.324382	−	0.945926i	$-0.605156\pi$
$743$	−13139.2	−0.648765	−0.324382	+	0.945926i	$0.605156\pi$
$744$	0	0
$745$	5033.92	0.247555
$746$	0	0
$747$	21930.2	1.07414
$748$	0	0
$749$	−12772.0	−0.623070
$750$	0	0
$751$	−27803.1	−1.35093	−0.675466	−	0.737391i	$-0.736058\pi$
$751$	−27803.1	−1.35093	−0.675466	+	0.737391i	$0.736058\pi$
$752$	0	0
$753$	−10982.1	−0.531490
$754$	0	0
$755$	9917.67	0.478068
$756$	0	0
$757$	26943.6	1.29364	0.646818	−	0.762644i	$-0.276099\pi$
$757$	26943.6	1.29364	0.646818	+	0.762644i	$0.276099\pi$
$758$	0	0
$759$	−4250.29	−0.203262
$760$	0	0
$761$	8418.10	0.400993	0.200497	−	0.979694i	$-0.435744\pi$
$761$	8418.10	0.400993	0.200497	+	0.979694i	$0.435744\pi$
$762$	0	0
$763$	13607.9	0.645663
$764$	0	0
$765$	−983.901	−0.0465007
$766$	0	0
$767$	8206.97	0.386358
$768$	0	0
$769$	−11445.7	−0.536726	−0.268363	−	0.963318i	$-0.586483\pi$
$769$	−11445.7	−0.536726	−0.268363	+	0.963318i	$0.586483\pi$
$770$	0	0
$771$	−17807.6	−0.831809
$772$	0	0
$773$	13064.8	0.607902	0.303951	−	0.952688i	$-0.401694\pi$
$773$	13064.8	0.607902	0.303951	+	0.952688i	$0.401694\pi$
$774$	0	0
$775$	−34214.6	−1.58584
$776$	0	0
$777$	−4368.08	−0.201678
$778$	0	0
$779$	29478.8	1.35583
$780$	0	0
$781$	1809.55	0.0829078
$782$	0	0
$783$	−33530.4	−1.53037
$784$	0	0
$785$	−8677.43	−0.394536
$786$	0	0
$787$	39876.6	1.80616	0.903080	−	0.429473i	$-0.141300\pi$
$787$	39876.6	1.80616	0.903080	+	0.429473i	$0.141300\pi$
$788$	0	0
$789$	3151.24	0.142189
$790$	0	0
$791$	2872.41	0.129116
$792$	0	0
$793$	15671.8	0.701792
$794$	0	0
$795$	−4483.17	−0.200002
$796$	0	0
$797$	27650.2	1.22888	0.614442	−	0.788962i	$-0.289381\pi$
$797$	27650.2	1.22888	0.614442	+	0.788962i	$0.289381\pi$
$798$	0	0
$799$	−10099.7	−0.447186
$800$	0	0
$801$	526.234	0.0232129
$802$	0	0
$803$	−13273.3	−0.583317
$804$	0	0
$805$	−2124.88	−0.0930338
$806$	0	0
$807$	−10573.3	−0.461213
$808$	0	0
$809$	4760.40	0.206881	0.103441	−	0.994636i	$-0.467015\pi$
$809$	4760.40	0.206881	0.103441	+	0.994636i	$0.467015\pi$
$810$	0	0
$811$	−34075.5	−1.47540	−0.737701	−	0.675127i	$-0.764089\pi$
$811$	−34075.5	−1.47540	−0.737701	+	0.675127i	$0.764089\pi$
$812$	0	0
$813$	13390.6	0.577648
$814$	0	0
$815$	3045.12	0.130878
$816$	0	0
$817$	15705.1	0.672524
$818$	0	0
$819$	8501.31	0.362711
$820$	0	0
$821$	−2245.59	−0.0954589	−0.0477294	−	0.998860i	$-0.515199\pi$
$821$	−2245.59	−0.0954589	−0.0477294	+	0.998860i	$0.515199\pi$
$822$	0	0
$823$	12914.3	0.546980	0.273490	−	0.961875i	$-0.411822\pi$
$823$	12914.3	0.546980	0.273490	+	0.961875i	$0.411822\pi$
$824$	0	0
$825$	8685.45	0.366532
$826$	0	0
$827$	−40085.3	−1.68549	−0.842747	−	0.538309i	$-0.819063\pi$
$827$	−40085.3	−1.68549	−0.842747	+	0.538309i	$0.819063\pi$
$828$	0	0
$829$	−3101.91	−0.129956	−0.0649782	−	0.997887i	$-0.520698\pi$
$829$	−3101.91	−0.129956	−0.0649782	+	0.997887i	$0.520698\pi$
$830$	0	0
$831$	−20994.7	−0.876410
$832$	0	0
$833$	3789.46	0.157619
$834$	0	0
$835$	7122.39	0.295186
$836$	0	0
$837$	−42874.9	−1.77058
$838$	0	0
$839$	46317.7	1.90592	0.952958	−	0.303103i	$-0.0980225\pi$
$839$	46317.7	1.90592	0.952958	+	0.303103i	$0.0980225\pi$
$840$	0	0
$841$	32059.0	1.31449
$842$	0	0
$843$	26237.6	1.07197
$844$	0	0
$845$	−94.7327	−0.00385669
$846$	0	0
$847$	−8407.84	−0.341082
$848$	0	0
$849$	27370.4	1.10642
$850$	0	0
$851$	−6763.34	−0.272438
$852$	0	0
$853$	32134.3	1.28987	0.644934	−	0.764238i	$-0.276885\pi$
$853$	32134.3	1.28987	0.644934	+	0.764238i	$0.276885\pi$
$854$	0	0
$855$	3916.79	0.156668
$856$	0	0
$857$	−11227.7	−0.447529	−0.223765	−	0.974643i	$-0.571835\pi$
$857$	−11227.7	−0.447529	−0.223765	+	0.974643i	$0.571835\pi$
$858$	0	0
$859$	−28714.1	−1.14053	−0.570263	−	0.821462i	$-0.693159\pi$
$859$	−28714.1	−1.14053	−0.570263	+	0.821462i	$0.693159\pi$
$860$	0	0
$861$	15504.6	0.613698
$862$	0	0
$863$	22544.0	0.889229	0.444615	−	0.895722i	$-0.353341\pi$
$863$	22544.0	0.889229	0.444615	+	0.895722i	$0.353341\pi$
$864$	0	0
$865$	−2335.50	−0.0918029
$866$	0	0
$867$	938.685	0.0367698
$868$	0	0
$869$	−23821.8	−0.929918
$870$	0	0
$871$	7809.19	0.303794
$872$	0	0
$873$	−26869.1	−1.04167
$874$	0	0
$875$	9161.62	0.353965
$876$	0	0
$877$	10.8851	0.000419114 0	0.000209557	−	1.00000i	$-0.499933\pi$
$877$	10.8851	0.000419114 0	0.000209557		1.00000i	$0.499933\pi$
$878$	0	0
$879$	−5660.75	−0.217215
$880$	0	0
$881$	5955.61	0.227752	0.113876	−	0.993495i	$-0.463673\pi$
$881$	5955.61	0.227752	0.113876	+	0.993495i	$0.463673\pi$
$882$	0	0
$883$	30192.6	1.15069	0.575347	−	0.817910i	$-0.304867\pi$
$883$	30192.6	1.15069	0.575347	+	0.817910i	$0.304867\pi$
$884$	0	0
$885$	1988.73	0.0755373
$886$	0	0
$887$	−5404.65	−0.204589	−0.102295	−	0.994754i	$-0.532618\pi$
$887$	−5404.65	−0.204589	−0.102295	+	0.994754i	$0.532618\pi$
$888$	0	0
$889$	7918.76	0.298748
$890$	0	0
$891$	338.013	0.0127092
$892$	0	0
$893$	40205.6	1.50664
$894$	0	0
$895$	−247.150	−0.00923051
$896$	0	0
$897$	−8441.71	−0.314226
$898$	0	0
$899$	72179.4	2.67777
$900$	0	0
$901$	−6669.30	−0.246600
$902$	0	0
$903$	8260.19	0.304410
$904$	0	0
$905$	3260.57	0.119762
$906$	0	0
$907$	−594.900	−0.0217788	−0.0108894	−	0.999941i	$-0.503466\pi$
$907$	−594.900	−0.0217788	−0.0108894	+	0.999941i	$0.503466\pi$
$908$	0	0
$909$	16774.9	0.612089
$910$	0	0
$911$	−16139.5	−0.586965	−0.293483	−	0.955964i	$-0.594814\pi$
$911$	−16139.5	−0.586965	−0.293483	+	0.955964i	$0.594814\pi$
$912$	0	0
$913$	31653.4	1.14740
$914$	0	0
$915$	3797.62	0.137208
$916$	0	0
$917$	18335.8	0.660306
$918$	0	0
$919$	−54241.0	−1.94695	−0.973474	−	0.228799i	$-0.926520\pi$
$919$	−54241.0	−1.94695	−0.973474	+	0.228799i	$0.926520\pi$
$920$	0	0
$921$	29550.1	1.05723
$922$	0	0
$923$	3594.05	0.128168
$924$	0	0
$925$	13820.9	0.491273
$926$	0	0
$927$	−12409.0	−0.439660
$928$	0	0
$929$	14010.7	0.494807	0.247404	−	0.968913i	$-0.420423\pi$
$929$	14010.7	0.494807	0.247404	+	0.968913i	$0.420423\pi$
$930$	0	0
$931$	−15085.4	−0.531044
$932$	0	0
$933$	27579.4	0.967746
$934$	0	0
$935$	−1420.13	−0.0496719
$936$	0	0
$937$	23701.6	0.826357	0.413178	−	0.910650i	$-0.364419\pi$
$937$	23701.6	0.826357	0.413178	+	0.910650i	$0.364419\pi$
$938$	0	0
$939$	437.571	0.0152072
$940$	0	0
$941$	−1871.24	−0.0648253	−0.0324126	−	0.999475i	$-0.510319\pi$
$941$	−1871.24	−0.0648253	−0.0324126	+	0.999475i	$0.510319\pi$
$942$	0	0
$943$	24006.6	0.829016
$944$	0	0
$945$	5441.27	0.187306
$946$	0	0
$947$	−51077.5	−1.75269	−0.876345	−	0.481684i	$-0.840025\pi$
$947$	−51077.5	−1.75269	−0.876345	+	0.481684i	$0.840025\pi$
$948$	0	0
$949$	−26362.7	−0.901759
$950$	0	0
$951$	9424.75	0.321365
$952$	0	0
$953$	9534.26	0.324076	0.162038	−	0.986784i	$-0.448193\pi$
$953$	9534.26	0.324076	0.162038	+	0.986784i	$0.448193\pi$
$954$	0	0
$955$	6933.21	0.234925
$956$	0	0
$957$	−18322.9	−0.618909
$958$	0	0
$959$	25626.3	0.862894
$960$	0	0
$961$	62504.1	2.09809
$962$	0	0
$963$	19172.4	0.641559
$964$	0	0
$965$	−7391.97	−0.246586
$966$	0	0
$967$	−8267.50	−0.274938	−0.137469	−	0.990506i	$-0.543897\pi$
$967$	−8267.50	−0.274938	−0.137469	+	0.990506i	$0.543897\pi$
$968$	0	0
$969$	−3736.79	−0.123883
$970$	0	0
$971$	−52149.6	−1.72354	−0.861771	−	0.507297i	$-0.830645\pi$
$971$	−52149.6	−1.72354	−0.861771	+	0.507297i	$0.830645\pi$
$972$	0	0
$973$	−2884.13	−0.0950266
$974$	0	0
$975$	17250.6	0.566627
$976$	0	0
$977$	−20321.3	−0.665441	−0.332721	−	0.943025i	$-0.607967\pi$
$977$	−20321.3	−0.665441	−0.332721	+	0.943025i	$0.607967\pi$
$978$	0	0
$979$	759.549	0.0247960
$980$	0	0
$981$	−20427.2	−0.664822
$982$	0	0
$983$	−15604.0	−0.506298	−0.253149	−	0.967427i	$-0.581466\pi$
$983$	−15604.0	−0.506298	−0.253149	+	0.967427i	$0.581466\pi$
$984$	0	0
$985$	−12919.1	−0.417906
$986$	0	0
$987$	21146.4	0.681963
$988$	0	0
$989$	12789.7	0.411213
$990$	0	0
$991$	22050.5	0.706818	0.353409	−	0.935469i	$-0.385022\pi$
$991$	22050.5	0.706818	0.353409	+	0.935469i	$0.385022\pi$
$992$	0	0
$993$	6157.74	0.196787
$994$	0	0
$995$	−4086.24	−0.130194
$996$	0	0
$997$	19850.1	0.630549	0.315275	−	0.949000i	$-0.397903\pi$
$997$	19850.1	0.630549	0.315275	+	0.949000i	$0.397903\pi$
$998$	0	0
$999$	17319.2	0.548503

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1088.4.a.bg.1.5		7
4.3	odd	2		1088.4.a.bh.1.3		7
8.3	odd	2		544.4.a.l.1.5	yes	7
8.5	even	2		544.4.a.k.1.3	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
544.4.a.k.1.3	✓	7	8.5	even	2
544.4.a.l.1.5	yes	7	8.3	odd	2
1088.4.a.bg.1.5		7	1.1	even	1	trivial
1088.4.a.bh.1.3		7	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1088 = 2^{6} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1088.a (trivial)

\(f(q)\)	\(=\)	\(q+3.24805 q^{3} -3.51829 q^{5} +10.9586 q^{7} -16.4502 q^{9} +O(q^{10})\)	\(q+3.24805 q^{3} -3.51829 q^{5} +10.9586 q^{7} -16.4502 q^{9} -23.7437 q^{11} -47.1585 q^{13} -11.4276 q^{15} -17.0000 q^{17} +67.6749 q^{19} +35.5940 q^{21} +55.1122 q^{23} -112.622 q^{25} -141.128 q^{27} +237.588 q^{29} +303.801 q^{31} -77.1206 q^{33} -38.5555 q^{35} -122.719 q^{37} -153.173 q^{39} +435.595 q^{41} +232.067 q^{43} +57.8765 q^{45} +594.099 q^{47} -222.909 q^{49} -55.2168 q^{51} +392.312 q^{53} +83.5372 q^{55} +219.811 q^{57} -174.029 q^{59} -332.321 q^{61} -180.271 q^{63} +165.917 q^{65} -165.594 q^{67} +179.007 q^{69} -76.2120 q^{71} +559.023 q^{73} -365.800 q^{75} -260.198 q^{77} +1003.29 q^{79} -14.2359 q^{81} -1333.13 q^{83} +59.8109 q^{85} +771.696 q^{87} -31.9895 q^{89} -516.791 q^{91} +986.760 q^{93} -238.100 q^{95} +1633.36 q^{97} +390.588 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 2 q^{7} + 67 q^{9}+O(q^{10}) \) 7 * q - 2 * q^7 + 67 * q^9	\( 7 q - 2 q^{7} + 67 q^{9} - 108 q^{11} + 34 q^{13} + 128 q^{15} - 119 q^{17} - 124 q^{19} + 296 q^{21} - 6 q^{23} + 197 q^{25} - 248 q^{29} + 50 q^{31} + 512 q^{33} - 640 q^{35} + 484 q^{37} + 1504 q^{39} - 366 q^{41} - 1412 q^{43} + 80 q^{45} + 1012 q^{47} + 1115 q^{49} + 146 q^{53} + 1024 q^{55} + 48 q^{57} - 2332 q^{59} + 548 q^{61} + 2838 q^{63} - 208 q^{65} - 924 q^{67} + 1672 q^{69} + 1286 q^{71} + 870 q^{73} - 3136 q^{75} + 1344 q^{77} + 1818 q^{79} + 3039 q^{81} - 1772 q^{83} + 384 q^{87} + 1706 q^{89} - 588 q^{91} + 5576 q^{93} + 2048 q^{95} + 1802 q^{97} - 5148 q^{99}+O(q^{100}) \) 7 * q - 2 * q^7 + 67 * q^9 - 108 * q^11 + 34 * q^13 + 128 * q^15 - 119 * q^17 - 124 * q^19 + 296 * q^21 - 6 * q^23 + 197 * q^25 - 248 * q^29 + 50 * q^31 + 512 * q^33 - 640 * q^35 + 484 * q^37 + 1504 * q^39 - 366 * q^41 - 1412 * q^43 + 80 * q^45 + 1012 * q^47 + 1115 * q^49 + 146 * q^53 + 1024 * q^55 + 48 * q^57 - 2332 * q^59 + 548 * q^61 + 2838 * q^63 - 208 * q^65 - 924 * q^67 + 1672 * q^69 + 1286 * q^71 + 870 * q^73 - 3136 * q^75 + 1344 * q^77 + 1818 * q^79 + 3039 * q^81 - 1772 * q^83 + 384 * q^87 + 1706 * q^89 - 588 * q^91 + 5576 * q^93 + 2048 * q^95 + 1802 * q^97 - 5148 * q^99

Introduction

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